For a \(m \times n\) array, i.e., for \(y \in \mathbb{R}^{m n}\), show that the Kronecker-product matrices

\[
J_{m} \otimes J_{n}, \quad J_{m} \otimes\left(I_{n}-J_{n}ight), \quad\left(I_{m}-J_{m}ight) \otimes J_{n}, \quad\left(I_{m}-J_{m}ight) \otimes\left(I_{n}-J_{n}ight)
\]

are complementary and mutually orthogonal projections \(\mathbb{R}^{m n} ightarrow \mathbb{R}^{m n}\) with ranks one, \(m-1, n-1\), and \((m-1)(n-1)\) respectively. Show also that this decomposition is invariant with respect to row and column permutation.